1. Solution
Week 8 (11/4/02)
Sub-rectangles
Put the rectangle in the x-y plane, with its sides parallel to the x and y axes,
and consider the function,
f(x, y) = e2πix
e2πiy
. (1)
Claim: The integral of f(x, y) over a rectangle, whose sides are parallel to the
axes, is zero if and only if at least one pair of sides has integer length.
Proof:
b
a
d
c
e2πix
e2πiy
dx dy =
b
a
e2πix
dx
d
c
e2πiy
dy (2)
= −
1
4π2
e2πib
− e2πia
e2πid
− e2πic
= −
e2πiae2πic
4π2
e2πi(b−a)
− 1 e2πi(d−c)
− 1 .
This equals zero if and only if at least one of the factors in parentheses is zero. In
other words, it equals zero if and only if at least one of (b − a) and (d − c) is an
integer.
Since we are told that each of the smaller rectangles has at least one integer pair
of sides, the integral of f over each of these smaller rectangles is zero. Therefore,
the integral of f over the whole rectangle is also zero. Hence, from the Claim, we
see that the whole rectangle has at least one integer pair of sides.
Remarks:
1. This same procedure works for the analogous problem in higher dimensions. Given
an N-dimensional rectangular parallelepiped which is divided into smaller ones, each
of which has the property that at least one edge has integer length, then the original
parallelepiped must also have this property. In three dimensions, for example, this
follows from considering integrals of the function f(x, y, z) = e2πix
e2πiy
e2πiz
, in the
same manner as above.
2. We can also be a bit more general and make the following statement. Given an
N-dimensional rectangular parallelepiped which is divided into smaller ones, each of
which has the property that at least n “non-equivalent” edges (that is, ones that
aren’t parallel) have integer lengths, then the original parallelepiped must also have
this property.
This follows from considering the N
m functions (where m ≡ N + 1 − n) of the form,
f(xj1 , xj2 , . . . , xjm ) = e2πixj1 e2πixj2 · · · e2πixjm , (3)
where the indices j1, j2, . . . jm are a subset of the indices 1, 2, . . . , N. (For example, the
previous remark deals with n = 1, m = N, and only N
N = 1 function.) We’ll leave it
to you to show that the integrals of these N
m functions, over an N-dimensional rect-
angular parallelepiped, are all equal to zero if and only if at least n “non-equivalent”
edges of the parallelepiped have integer length.